**Measure of a set** . Denote a *measure of an interval (a, b)* , where a <b is its length ba.

Let H be an open set bounded by the line. There is a proposition that H is equal to the countable union of open intervals two to two disjoint, any interval is (a _{j} , b _{j} ) .It is called the *measure m (H) of the bounded open set H* the sum of the lengths of its intervals (b _{j} – a _{j} .

It should be noted that if the number of the intervals (a _{j} , b _{j} ) is countable, then the sum of the lengths of the intervals is a numerical series, with positive terms (b _{j} – a _{j} ). Because H is a bounded set, this series is convergent.

## Case of an arbitrary set

Let F be an arbitrary bounded set. All sorts of open sets H containing F are analyzed. The set {mH} of the measures of these sets is bounded inferiorly, ie 0, and therefore has the smallest, inf {mH}.

The number m ^{*} F = inf {mH} is called the *outer measure* of the set F.